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%% cosmology

\newcommand{\Ob}{\Omega_{\rm b} }
\newcommand{\Om}{\Omega_{\rm m} }
\newcommand{\OHI}{\Omega_{\rm HI} }
\newcommand{\OHH}{\Omega_{\rm H2} }
\newcommand{\Ol}{\Omega_{\Lambda} }
\newcommand{\Mpch}{h^{-1} \rm{\,Mpc} }

\newcommand{\rhox}{\rho({\bf x})}
\newcommand{\rhomean}{\langle\rho\rangle}

\newcommand{\deltax}{\delta({\bf x})}
\newcommand{\deltak}{\tilde{\delta}({\bf k})}

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\newcommand{\mnras}{MNRAS}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\title{Cosmology}
\author{Gabriel Altay}
\begin{document}
\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
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These notes are meant to provide theoretical background for setting up cosmological simulations. 

\section{Correlation Function and Power Spectrum}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Fluctuations in the cosmological density field are often described using the density contrast, 
%
\begin{equation}
\deltax = \frac{\rhox - \rhomean}{ \rhomean }
\label{eqn:dc}
\end{equation}
%
where $\rhox$ is the matter density at a point in space ${\bf x}$ and $\rhomean$ is the mean matter density in the universe.  The density contrast is dimensionless and we can define a fourier transform pair as,
%
\begin{equation}
\deltak = \int d^3 {\bf x} \,  
\deltax e^{-i {\bf k} \cdot {\bf x}}, \quad
\deltax = \frac{1}{(2\pi)^3} \int d^3 {\bf k} \,  
\deltak e^{i {\bf k} \cdot {\bf x}}
\label{eqn:ft_pair}
\end{equation}
%
The factor of $1/(2\pi)^3$ represents a choice of fourier transform convention and the fact that $\deltax$ is dimensionless implies that $\deltak$ has dimensions of volume.  A common statistic used to quantify large scale structure is the two point correlation function, $\xi$.  One way to define the two point correlation function, $\xi$, is as the expectation value (or classical average over an ensemble) of the product of the density contrast at two points in space, $\deltax$ and $\delta({\bf x'})$.  Under the assumptions of homogeneity and isotropy (i.e. the cosmological principle), $\xi$ can only depend on the magnitude of the seperation, $r = |{\bf x-x'}|$. 
%
\begin{equation}
\xi({\bf x},{\bf x'}) = 
\langle \delta^*({\bf x}) \delta({\bf x'}) \rangle = 
\xi(r)
\label{eqn:cf_def}
\end{equation}
%
The complex conjugate of $\deltax$ is $\delta^*({\bf x})$ but in the case of the density contrast (which is real valued) they are equal.  In the the special case of $r=0$, $\xi(0)$ is the mass variance of the density field.  Likewise we can define the power spectrum, $P(k)$, as the ensemble average of the product of the fourier components, 
%
\begin{equation} 
\langle \tilde{\delta}^*({\bf k}) \tilde{\delta}({\bf k'}) \rangle =
P(k) \delta^D({\bf k - k'})
\label{eqn:ps_def} 
\end{equation}
%
where $\delta^D$ is the dirac delta function.  The dirac delta function has units that are the reciprocal of its argument implying that $P(k)$ has units of volume.  It is often convenient to define a dimensionless power spectrum as,
%
\begin{equation} 
\Delta^2_{\mathcal R}(k) = 
\frac{k^3}{2 \pi^2}  P(k)
\label{eqn:ups_def}  
\end{equation}
% 
The factor of $k^3$ ensures that $\Delta^2_{\mathcal R}(k)$ is unitless while the factor $1/(2 \pi^2)$ is an (arbitrary) convention which we have chosen to match the usage in the WMAP papers.  By combining Eqs. \ref{eqn:ft_pair}, \ref{eqn:cf_def}, and \ref{eqn:ps_def}, we can relate the power spectrum and the two point correlation. 
%
\begin{equation} 
\xi(r) = \int \frac{dk}{k} 
\Delta^2_{\mathcal R}(k)
j_0( k r ), \quad \text{where} \;
j_0(kr) = \frac{\sin(kr)}{kr}
\label{eqn:cf_ps_rel}  
\end{equation}
% 










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